Question

Suppose the samples x₁, ..., xₙ have the z-scores z₁, ..., zₙ.

(a) Show that Σᵢzᵢ = 0.
(b) Show that Σᵢzᵢ² = n-1."

Answer 1

The sum of z-scores is zero, and the sum of the squares of z-scores is n-1.

Step-by-step explanation:

To show that Σᵢzᵢ = 0, we can use the fact that the mean of the z-scores is zero. The mean of a set of values is the sum of the values divided by the number of values. Since all the z-scores have a mean of zero, their sum is also zero.

To show that Σᵢzᵢ² = n-1, we can use the fact that the variance of the z-scores is equal to 1. The variance of a set of values is the sum of the squared differences from the mean divided by the number of values minus 1. Since the variance of the z-scores is 1, their sum of squares is n-1.